2-1.The pseudopotential method

08-11

來自專欄 A note on DFT

Choosing $phi_{vec{K}}^{vec{k}}(vec{r})equiv|vec{K}>equiv e^{i(vec{k}+vec{K})cdotvec{r}}$ as basis functions, we can write the state as

$egin{eqnarray} psi_{vec{k}}^n(vec{r})=sum_{vec{K}}c^{n,vec{k}}_{vec{K}}|vec{K}> end{eqnarray}$

In practice we cannot work with an infinite basis set, and have to limit it somehow. than plane waves, we can easily limit the set to all $vec{K}$ with $|vec{K}|leq K_{max}$ , and the related cut-off energy is

$egin{eqnarray} E_{cut}equivfrac{hbar^2K^2_{max}}{2m_e} end{eqnarray}$

Plane waves are orthogonal:

$egin{eqnarray} <vec{K}_1|vec{K}_2>&equiv&int dvec{r}e^{i(vec{K}_2-vec{K}_1)cdotvec{r}} onumber\ &=&delta(vec{K}_2-vec{K}_1) end{eqnarray}$